Some examples of nontrivial homotopy groups of modules

C Joanna Su

doi:10.1155/s0161171201005373

Abstract

The concept of the homotopy theory of modules was discovered by Peter Hilton as a result of his trip in 1955 to Warsaw, Poland, to work with Karol Borsuk, and to Zurich, Switzerland, to work with Beno Eckmann. The idea was to produce an analog of homotopy theory in topology. Yet, unlike homotopy theory in topology, there are two homotopy theories of modules, the injective theory, , and the projective theory, . They are dual, but not isomorphic. In this paper, we deliver and carry out the precise calculation of the first known nontrivial examples of absolute homotopy groups of modules, namely, , and , where ℚ/ℤ and ℤ are regarded as ℤCk‐modules with trivial action. One interesting phenomenon of the results is the periodicity of these homotopy groups, just as for the Ext groups.

Highlights

It is well known that ExtnΛ(A, B) may be regarded either as the value of the left nth derived functor of HomΛ(−, B) on the Λ-module A or as the value of the right nth derived functor of HomΛ(A, −) on the Λ-module B
We study the injective homotopy groups π n(A, B) and the projective homotopy groups π n(A, B) which are, respectively, the value of the right nth derived functor of HomΛ(−, B) on the Λ-module A and the value of the left nth derived functor of HomΛ(A, −) on the Λmodule B
We refer to these abelian groups as homotopy groups because of their strong analogy with the homotopy groups encountered in the homotopy theory of pointed topological spaces

Summary

Introduction

It is well known that ExtnΛ(A, B) may be regarded either as the value of the left nth derived functor of HomΛ(−, B) on the Λ-module A or as the value of the right nth derived functor of HomΛ(A, −) on the Λ-module B. Mainly using the long exact (π , ExtΛ)-sequence in the first variable (see Theorem 2.7), we obtain another nontrivial example, namely, π n(Z, Q/Z), where Z is regarded as a trivial Ck-module. 2. Examples of nontrivial injective homotopy groups of modules. Let the abelian groups D and B be regarded as trivial Ck-modules.

Results

Conclusion